Uniform $L^{\infty}$-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains

Abstract

We consider a family of semilinear parabolic equations with homogeneous Neumann boundary conditions on a family of varying non-smooth domains $\{\Omega_\mu\}_{\mu \in \Lambda} \subset \mathbb{R}^n$. Assuming only that the domains have uniformly bounded volumes, satisfy a uniform Jones condition, and possess uniform ellipticity bounds, we establish the well-posedness of the problem in an appropriate scale of fractional Banach spaces and prove the existence of global attractors. Using a Moser-Alikakos bootstrap iteration in tandem with the uniform Gronwall lemma and the uniform properties of the Jones extension operator, we show that the family of attractors is uniformly bounded in $L^\infty(\Omega_\mu)$. Finally, assuming the volume convergence of the domains, $|\Omega_\mu \triangle \Omega_0| \to 0$, we construct a framework of connecting maps to prove that the family of attractors is upper semicontinuous at $\mu = 0$ in the strong $H^1$ topology.